Question

Evaluate the following. 7. \( \sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^{2 n} \) 8. \( \sum_{n=0}^{\infty} \frac{(\ln 4)^{n}}{2^{n} n!} \) \( \frac{\left(\frac{1 n 4}{2}\right)^{n}}{x^{n} n!} \) 9. \( \sum_{n=0}^{\infty} \frac{(-1)^{n}(\pi)^{2 n+1}}{3^{2 n+1}(2 n+1)!} \) \[ \begin{aligned} & =\frac{x^{n} n!}{n!} \\ \sum_{n=0}^{\infty} \frac{x^{n}}{n!} & =e^{x} \rightarrow e^{\frac{\ln 4}{2}} \rightarrow e^{\ln 4^{1 / 2}} \\ & =2 \end{aligned} \] 10. \( \sum_{n=0}^{\infty} \frac{3^{n}}{5^{n+1} n!} \) 11. \( \sum_{n=0}^{\infty} \frac{(-1)^{n}(\pi)^{2 n}}{3^{2 n}(2 n)!}=\frac{(-1)^{n} \times x^{2 n}}{(2 n)!}=\cos x \) 12. \( \sum_{n=0}^{\infty} \frac{(-1)^{n}(3)^{\frac{2 n+1}{2}}}{(2 n+1)} \) \[ \begin{array}{l} \operatorname{cor} x=\frac{\pi}{3} \\ \cos (\pi / 3)=1 / 2 \end{array} \] 13. \( \sum_{n=0}^{\infty} \frac{(-1)^{n}(\pi)^{2 n}}{(2 n)!} \) 14. \( 1-\ln 3+\frac{(\ln 3)^{2}}{2!}-\frac{(\ln 3)^{3}}{3!}+\ldots \) 15. \( 3-e+\frac{e^{2}}{3}-\frac{e^{3}}{3^{2}}+\ldots \)

          Evaluate the following.
7. \( \sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^{2 n} \)
8. \( \sum_{n=0}^{\infty} \frac{(\ln 4)^{n}}{2^{n} n!} \)
\( \frac{\left(\frac{1 n 4}{2}\right)^{n}}{x^{n} n!} \)
9. \( \sum_{n=0}^{\infty} \frac{(-1)^{n}(\pi)^{2 n+1}}{3^{2 n+1}(2 n+1)!} \)
\[
\begin{aligned}
& =\frac{x^{n} n!}{n!} \\
\sum_{n=0}^{\infty} \frac{x^{n}}{n!} & =e^{x} \rightarrow e^{\frac{\ln 4}{2}} \rightarrow e^{\ln 4^{1 / 2}} \\
& =2
\end{aligned}
\]
10. \( \sum_{n=0}^{\infty} \frac{3^{n}}{5^{n+1} n!} \)
11. \( \sum_{n=0}^{\infty} \frac{(-1)^{n}(\pi)^{2 n}}{3^{2 n}(2 n)!}=\frac{(-1)^{n} \times x^{2 n}}{(2 n)!}=\cos x \)
12. \( \sum_{n=0}^{\infty} \frac{(-1)^{n}(3)^{\frac{2 n+1}{2}}}{(2 n+1)} \)
\[
\begin{array}{l}
\operatorname{cor} x=\frac{\pi}{3} \\
\cos (\pi / 3)=1 / 2
\end{array}
\]
13. \( \sum_{n=0}^{\infty} \frac{(-1)^{n}(\pi)^{2 n}}{(2 n)!} \)
14. \( 1-\ln 3+\frac{(\ln 3)^{2}}{2!}-\frac{(\ln 3)^{3}}{3!}+\ldots \)
15. \( 3-e+\frac{e^{2}}{3}-\frac{e^{3}}{3^{2}}+\ldots \)

Evaluate the following.
7. ∑n=1^∞((1)/(2))^2 n
8. ∑n=0^∞((ln 4)^n)/(2^n n!)
(((1 n 4)/(2))^n)/(x^n n!)
9. ∑n=0^∞((-1)^n(π)^2 n+1)/(3^2 n+1(2 n+1)!)

=(x^n n!)/(n!)
∑n=0^∞(x^n)/(n!) =e^x→ e^(ln 4)/(2)→ e^ln 4^1 / 2
=2

10. ∑n=0^∞(3^n)/(5^n+1 n!)
11. ∑n=0^∞((-1)^n(π)^2 n)/(3^2 n(2 n)!)=((-1)^n× x^2 n)/((2 n)!)=cos x
12. ∑n=0^∞((-1)^n(3)^(2 n+1)/(2))/((2 n+1))

cor x=(π)/(3)
cos (π / 3)=1 / 2

13. ∑n=0^∞((-1)^n(π)^2 n)/((2 n)!)
14. 1-ln 3+((ln 3)^2)/(2!)-((ln 3)^3)/(3!)+…
15. 3-e+(e^2)/(3)-(e^3)/(3^2)+…

Added by Stephen C.

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